Why is the rotation gizmo not working in 2.74?
It's not a bug, you can imagine this widget is something that's used by an incredible amount of people and is heavily tested. But this issue has cropped up from time to time and depends heavily on the graphics card drivers used. 

Try Setting "Selection" in User Preferences System Tab from "Auto" to "OpenGL Select" or "OpenGL Occlusion Queries"

The keyboard shortcuts for the transform tools are G, (grab), R (rotate), and S (scale), often they are more convenient, in combination with their follow up keys x, y, z to select the axis..